PSI - Issue 22

N. Makhutov et al. / Procedia Structural Integrity 22 (2019) 93–101 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

96

4

Similar concepts of pseudoplastic (fictitious) stresses σ p-f and strain ε p-f may also be introduced for approximation of notch stresses and strains when nominally plastic behavior according equation (7) is assumed:

K





t n 

p f 

n Y   

,

for

(10)

K





t n 

p f 

Processes of fracture of structural components are usually initiated in these highly stressed zones (Makhutov, 2008; Makhutov 2019). Assessment of the material response to various loading regimes in these zones is thus one of the key problems in ensuring structural integrity and safety. Such assessment should be carried out not only for the cases of normal loading regimes that cause elastic (ε<ε Y ) and limited elasto-plastic (ε<ε lp =5ε Y ) response of the material in the notch zone (regions I and II of the stress-strain curve, fig.2), but also for the cases of extreme loading that causes extensive plastic strains and general yielding of the cross section (region III fig.2 when maximum local strains tend to fracture values that can reach up to ε f ~50  70% or 20ε Y ). The assessment of stress strain response of the material at the notch zone in post-yielding situation is a complicated task. Closed form solutions are only available for a relatively small number of specific cases. Three types of approaches are used in notch mechanics: experimental strain measurements, numerical simulations (FEM), and approximate analytical methods also known as stress-strain conversion rules. Approximate analytical methods for assessment of stress-strain response in the notch zone are based on the following two types of equations: (1) Constitutive laws that relate maximum local stresses and strains ( σ max , ε max ) in the stress concentration zone in the plastic region. For the case of monotonic loading a Ramberg-Osgud relation might be used:   1/ max max max m E K      ; (11) (2) Postulated expressions that relate the values of elastic pseudo stresses and strains that are calculated by elasticity theory in the assumption that material in the notch zone is deformed elastically, and the actual elasto plastic response of the material at the notch root ( Adibi-Asl and Seshadri , 2009). max max { , } { , } e f e f         for , e f Y e f Y         . (12) The transformation Φ in essence determines the mapping of points A e-f ( σ e-f ; ε e-f ) of the semiaxis CE ∞ of the fictitious elastic pseudo states of the material to points A mak (σ max ;ε max ) of the segment CP ∞ of the stress-strain curve (fig.3) ( Makhutov and Reznikov, 2018) : (13) A number of approximate analytical methods of that kind (which are commonly known as stress-strain conversion rules) were developed and are widely applied along with numerical and experimental ones. These include: Linear rule, Neuber rule, Equivalent Strain Energy Density method and others. Neuber rule proved to be the most convenient and widely used approximate analytical method allowing determination of the stress-strain material response at the notch zone in elastoplastic formulation. It relates stress and strain concentration factors ( K σ and K ε ) to the theoretical stress concentration factor K t (Neuber, H., 1961): max max max   A ( e f  , e f e f     ) A ( , )   for ,     e f  Y e f  Y   .

2 K K K

1    ,

(14)

t

It can be also rewritten in the form that directly relates maximum local stresses and strains to (fictitious) pseudoelastic stresses and strains that are calculated using the elasticity theory in the assumption that the material at the notch root remains linear elastic when local stresses exceed the yield strength:

max max e f e f          .

(15)